Divisibility sequence

In mathematics, a divisibility sequence is an integer sequence $$(a_n)$$ indexed by positive integers n such that


 * $$\text{if }m\mid n\text{ then }a_m\mid a_n$$

for all m, n. That is, whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined.

A strong divisibility sequence is an integer sequence $$(a_n)$$ such that for all positive integers m, n,


 * $$\gcd(a_m,a_n) = a_{\gcd(m,n)}.$$

Every strong divisibility sequence is a divisibility sequence: $$\gcd(m,n) = m$$ if and only if $$m\mid n$$. Therefore, by the strong divisibility property, $$\gcd(a_m,a_n) = a_m$$ and therefore $$a_m\mid a_n$$.

Examples

 * Any constant sequence is a strong divisibility sequence.
 * Every sequence of the form $$a_n = kn,$$ for some nonzero integer k, is a divisibility sequence.
 * The numbers of the form $$2^n-1$$ (Mersenne numbers) form a strong divisibility sequence.
 * The repunit numbers in any base $R_{n}^{(b)}$ form a strong divisibility sequence.
 * More generally, any sequence of the form $$a_n = A^n - B^n$$ for integers $$A>B>0$$ is a divisibility sequence. In fact, if $$A$$ and $$B$$ are coprime, then this is a strong divisibility sequence.
 * The Fibonacci numbers $F_{n}$ form a strong divisibility sequence.
 * More generally, any Lucas sequence of the first kind $U_{n}(P,Q)$ is a divisibility sequence. Moreover, it is a strong divisibility sequence when $gcd(P,Q) = 1$.
 * Elliptic divisibility sequences are another class of such sequences.